Heuristic syllogism and Nature of plausible reasoning

I found the following discussion on heuristic syllogism and the nature of plausible reasoning interesting, and would like to share it with you. It is quite long, but worth reading. It is taken from the book How to Solve It: A New Aspect of Mathematical Method by George Polya. And perhaps you can apply it to the recent discussion of the lost tomb.

Heuristic syllogism. We begin by restating a mode of heuristic reasoning in the following form:

If we are approaching land, we often see birds.
Now we see birds.
Therefore, it becomes more credible that we are approaching land.

The two statements above the horizontal line may be called the premises, the statement under the line, the conclusion. And the whole pattern of reasoning may be termed a heuristic syllogism.

An essential circumstance is better emphasized. Columbus and his men conjectured from the beginning that they would eventually find land sailing westward; and otherwise they would not have started out at all. As they proceeded, they related every incident, major or minor, to their dominating question: “Are we approaching land?” Their confidence rose and fell as events occurred or failed to occur, and each man’s beliefs fluctuated more or less differently according to his background and character. The whole dramatic tension of the voyage is due to such fluctuations of confidence.

The heuristic syllogism quoted exhibits a reasonable ground for a change in the level of confidence. To occasion such changes is the essential role of this kind of reasoning.

The general pattern suggested by our example can be exhibited thus:

If A is true, then B is also true, as we know.
Now, it turns out that B is true.
Therefore, A becomes more credible.

Still shorter:

If A then B
B true
A more credible

In this schematic statement the horizontal line stands for the word “therefore” and expresses the implication, the essential link between the premises and the conclusion.

Nature of plausible reasoning. In this little book we are discussing a philosophical question. We discuss it as practically and informally and as far from high-brow modes of expression as we can, but nevertheless our subject is philosophical. It is concerned with the nature of heuristic reasoning and, by extension, with a kind of reasoning which is nondemonstrative although important and which we shall call, for lack of a better term, plausible reasoning.

The signs that convince the inventor that his idea is good, the indications that guide us in our everuday affairs, the circumstantial evidence of the lawyer, the inductive evidence of the scientist, statistical evidence invoked in many and diverse subjects—all these kinds of evidence agree in two essential points. First, they do not have the certainty of a strict demonstration. Second, they are useful in acquiring essentially new knowledge, and even indispensable to any not purely mathematical or logical knowledge, to any knowledge concerned with the physical world. We could call the reasoning that underlies this kind of evidence “heuristic reasoning” or “inductive reasoning” or (if we wish to avoid stretching the meaning of existing terms) “plausible reasoning.” We accept here the last term.

The heuristic syllogism introduced in the foregoing may be regarded as the simplest and most widespread pattern of plausible reasoning. It reminds us of a classical pattern of demonstrative reasoning, of the so-called “modus tollens of hypothetical syllogism.” We exhibit here both patterns:

If A then B
B false
A false

If A then B
B true
A more credible

The comparison of these patterns may be instructive. It may grant us an insight, not easily obtainable elsewhere, into the nature of plausible (heuristic, inductive) reasoning.

Both patterns have the same first premise:

If A then B.

They differ in the second premise. The statements:

B false

B true

are exactly opposite to each other but they are of “similar logical nature,” they are on the same “logical level.” The great difference arises after the premises. The conclusions

A false

A more credible

are on different logical levels and their relations to their respective premises are of a different logical nature.

The conclusion of the demonstrative syllogism is of the same logical nature as the premises. Moreover, this conclusion is fully expressed and is fully supported by the premises. If my neighbor and I agree to accept the premises, we cannot reasonably disagree about accepting also the conclusion, however different our tastes or other convictions may be.

The conclusion of the heuristic syllogism differs from the premises in its logical nature; it is more vague, not so sharp, less fully expressed. This conclusion is comparable to a force, has direction and magnitude. It pushes us in a certain direction: A becomes more credible. The conclusion also has a certain strength: A may become much more credible, or just a little more credible. The conclusion is not fully expressed and is not fully supported by the premises. The direction is expressed and is implied by the premises, the magnitude is not. For any reasonable person, the premises involve that A becomes more credible (certainly not less credible). Yet my neighbor and I can honestly disagree how much more credible A becomes, since our temperaments, our backgrounds, and our unstated reasons may be different.

In the demonstrative syllogism the premises constitute a full basis on which the conclusion rests. If both premises stand, the conclusion stands too. If we receive some new information that does not change our belief in the premises, it cannot change our belief in the conclusion.

In the heuristic syllogism the premises constitute only one part of the basis on which the conclusion rests, the fully expressed, the “visible” part of the basis; there is an unexpressed, invisible part, formed by something else, by inarticulate feelings perhaps, or by unstated reasons. In fact, it can happen that we receive some new information that leaves our belief in both premises completely intact, but influences the trust we put in A in a way just opposite to that expressed in the conclusion. To find A more plausible on the ground of the premises of our heuristic syllogism is only reasonable. Yet tomorrow I may find grounds, not interfering at all with these premises, that make A appear less plausible, or even definitively refute it. The conclusion may be shaken and even overturned completely by commotions in the invisible parts of its foundation, although the premises, the visible part, stand quite firm.

These remarks seem to make somewhat more understandable the nature of heuristic, inductive, and other sorts of not demonstrative plausible reasoning, which appear so baffling and elusive when approached from the point of view of purely demonstrative logic.

Heuristic reasons are important although they prove nothing. To clarify our heuristic reasons is also important although behind any reason clarified there are many others that remain obscure and are perhaps still more important.

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